Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8: Difference between revisions

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<td rowspan="3"><math>5</math></td>
<td rowspan="3"><math>5</math></td>
<td>1.1</td>
<td>1.1</td>
<td><math>x^3</math></td>
<td><span class="htmlMath">x<sup>3</sup></span></td>
<td>330</td>
<td>330</td>
<td>42</td>
<td>42</td>
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<tr>
<tr>
<td>1.2</td>
<td>1.2</td>
<td><math>x^5</math></td>
<td><span class="htmlMath">x<sup>5</sup></span></td>
<td>330</td>
<td>330</td>
<td>42</td>
<td>42</td>
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<tr>
<tr>
<td>2.1</td>
<td>2.1</td>
<td><math>x^{-1}</math></td>
<td><span class="htmlMath">x<sup>-1</sup></span></td>
<td>496</td>
<td>496</td>
<td>232</td>
<td>232</td>
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<td rowspan="14"><math>6</math></td>
<td rowspan="14"><math>6</math></td>
<td>1.1</td>
<td>1.1</td>
<td><math>x^{3}</math></td>
<td><span class="htmlMath">x<sup>3</sup></span></td>
<td>1102</td>
<td>1102</td>
<td>94</td>
<td>94</td>
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<tr>
<tr>
<td>1.2</td>
<td>1.2</td>
<td><math>x^{3} + u^{11}x^{6} + ux^{9}</math></td>
<td><span class="htmlMath">x<sup>3</sup> + u<sup>11</sup>x<sup>6</sup> + ux<sup>9</sup></span></td>
<td>1146</td>
<td>1146</td>
<td>94</td>
<td>94</td>
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<tr>
<tr>
<td>2.1</td>
<td>2.1</td>
<td><math>ux^{5} + x^{9} + u^{4}x^{17} + ux^{18} + u^{4}x^{20} + ux^{24} + u^{4}x^{34} + ux^{40}</math></td>
<td><span class="htmlMath">ux<sup>5</sup> + x<sup>9</sup> + u<sup>4</sup>x<sup>17</sup> + ux<sup>18</sup> + u<sup>4</sup>x<sup>20</sup> + ux<sup>24</sup> + u<sup>4</sup>x<sup>34</sup> + ux<sup>40</sup></span></td>
<td>1158</td>
<td>1158</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.2</td>
<td>2.2</td>
<td><math>u^{7}x^{3} + x^{5} + u^{3}x^{9} + u^{4}x^{10} + x^{17} + u^{6}x^{18}</math></td>
<td><span class="htmlMath">u<sup>7</sup>x<sup>3</sup> + x<sup>5</sup> + u<sup>3</sup>x<sup>9</sup> + u<sup>4</sup>x<sup>10</sup> + x<sup>17</sup> + u<sup>6</sup>x<sup>18</sup></span></td>
<td>1166</td>
<td>1166</td>
<td>94</td>
<td>94</td>
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<tr>
<tr>
<td>2.3</td>
<td>2.3</td>
<td><math>x^{3} + ux^{24} + x^{10}</math></td>
<td><span class="htmlMath">x<sup>3</sup> + ux<sup>24</sup> + x<sup>10</sup></span></td>
<td>1166</td>
<td>1166</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.4</td>
<td>2.4</td>
<td><math>x^{3} + u^{17}(x^{17} + x^{18} + x^{20} + x^{24})</math></td>
<td><span class="htmlMath">x<sup>3</sup> + u<sup>17</sup>(x<sup>17</sup> + x<sup>18</sup> + x<sup>20</sup> + x<sup>24</sup>)</span></td>
<td>1168</td>
<td>1168</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.5</td>
<td>2.5</td>
<td><math>x^{3} + u^{11}x^{5} + u^{13}x^{9} + x^{17} + u^{11}x^{33} + x^{48}</math></td>
<td><span class="htmlMath">x<sup>3</sup> + u<sup>11</sup>x<sup>5</sup> + u<sup>13</sup>x<sup>9</sup> + x<sup>17</sup> + u<sup>11</sup>x<sup>33</sup> + x<sup>48</sup></span></td>
<td>1170</td>
<td>1170</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.6</td>
<td>2.6</td>
<td><math>u^{25}x^{5} + x^{9} + u^{38}x^{12} + u^{25}x^{18} + u^{25}x^{36}</math></td>
<td><span class="htmlMath">u<sup>25</sup>x<sup>5</sup> + x<sup>9</sup> + u<sup>38</sup>x<sup>12</sup> + u<sup>25</sup>x<sup>18</sup> + u<sup>25</sup>x<sup>36</sup></span></td>
<td>1170</td>
<td>1170</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.7</td>
<td>2.7</td>
<td><math>u^{40}x^{5} + u^{10}x^{6} + u^{62}x^{20} + u^{35}x^{33} + u^{15}x^{34} + u^{29}x^{48}</math></td>
<td><span class="htmlMath">u<sup>40</sup>x<sup>5</sup> + u<sup>10</sup>x<sup>6</sup> + u<sup>62</sup>x<sup>20</sup> + u<sup>35</sup>x<sup>33</sup> + u<sup>15</sup>x<sup>34</sup> + u<sup>29</sup>x<sup>48</sup></span></td>
<td>1170</td>
<td>1170</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.8</td>
<td>2.8</td>
<td><math>u^{34}x^{6} + u^{52}x^{9} + u^{48}x^{12} + u^{6}x^{20} + u^{9}x^{33} + u^{23}x^{34} + u^{25}x^{40}</math></td>
<td><span class="htmlMath">u<sup>34</sup>x<sup>6</sup> + u<sup>52</sup>x<sup>9</sup> + u<sup>48</sup>x<sup>12</sup> + u<sup>6</sup>x<sup>20</sup> + u<sup>9</sup>x<sup>33</sup> + u<sup>23</sup>x<sup>34</sup> + u<sup>25</sup>x<sup>40</sup></span></td>
<td>1170</td>
<td>1170</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.9</td>
<td>2.9</td>
<td><math>x^{9} + u^{4}(x^{10} + x^{18}) + u^{9}(x^{12} + x^{20} + x^{40})</math></td>
<td><span class="htmlMath">x<sup>9</sup> + u<sup>4</sup>(x<sup>10</sup> + x<sup>18</sup>) + u<sup>9</sup>(x<sup>12</sup> + x<sup>20</sup> + x<sup>40</sup>)</span></td>
<td>1172</td>
<td>1172</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.10</td>
<td>2.10</td>
<td><math>u^{52}x^{3} + u^{47}x^{5} + ux^{6} + u^{9}x^{9} + u^{44}x^{12} + u^{47}x^{33} + u^{10}x^{34} + u^{33}x^{40}</math></td>
<td><span class="htmlMath">u<sup>52</sup>x<sup>3</sup> + u<sup>47</sup>x<sup>5</sup> + ux<sup>6</sup> + u<sup>9</sup>x<sup>9</sup> + u<sup>44</sup>x<sup>12</sup> + u<sup>47</sup>x<sup>33</sup> + u<sup>10</sup>x<sup>34</sup> + u<sup>33</sup>x<sup>40</sup></span></td>
<td>1172</td>
<td>1172</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.11</td>
<td>2.11</td>
<td><math>u(x^{6} + x^{10} + x^{24} + x^{33}) + x^{9} + u^{4}x^{17}</math></td>
<td><span class="htmlMath">u(x<sup>6</sup> + x<sup>10</sup> + x<sup>24</sup> + x<sup>33</sup>) + x<sup>9</sup> + u<sup>4</sup>x<sup>17</sup></span></td>
<td>1174</td>
<td>1174</td>
<td>96</td>
<td>96</td>
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<tr>
<tr>
<td>2.12</td>
<td>2.12</td>
<td><math>x^{3} + </math> <math>u^{17}(x^{17} + </math> <math>x^{18} + </math> <math>x^{20} + </math> <math>x^{24}) + </math> <math>u^{14}((u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{2} + </math> <math>(u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{4}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{8}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{16}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{32}+ (u^{2}x)^{9} +(u^{2}x)^{19} +(u^{2}x)^{36} + </math> <math>x^{21}+x^{42}</math></td>
<td><span class="htmlMath">x<sup>3</sup> + u<sup>17</sup>(x<sup>17</sup> + x<sup>18</sup> + x<sup>20</sup> + x<sup>24</sup>) + u<sup>14</sup>((u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)+ (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>2</sup> + (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>4</sup>+ (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>8</sup>+ (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>16</sup>+ (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>32</sup>+ (u<sup>2</sup>x)<sup>9</sup> +(u<sup>2</sup>x)<sup>19</sup> +(u<sup>2</sup>x)<sup>36</sup> + x<sup>21</sup>+x<sup>42</sup></span></td>
<td>1300</td>
<td>1300</td>
<td>152</td>
<td>152</td>

Revision as of 17:10, 11 July 2020

Known switching classes of APN functions over [math]\displaystyle{ \mathbb{F}_{2^5} }[/math], [math]\displaystyle{ \mathbb{F}_{2^6} }[/math], [math]\displaystyle{ \mathbb{F}_{2^7} }[/math] and [math]\displaystyle{ \mathbb{F}_{2^8} }[/math].

Also available is Magma code generating representatives from the switching classes.

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] [math]\displaystyle{ F(x) }[/math] Γ-rank Δ-rank Aut(dev(GF))/22n Aut(dev(GF))/22n
[math]\displaystyle{ 5 }[/math] 1.1 x3 330 42 4960 4960
1.2 x5 330 42 4960 158720
2.1 x-1 496 232 310 310
[math]\displaystyle{ 6 }[/math] 1.1 x3 1102 94 24192 48384
1.2 x3 + u11x6 + ux9 1146 94 4032 8064
2.1 ux5 + x9 + u4x17 + ux18 + u4x20 + ux24 + u4x34 + ux40 1158 96 320 320
2.2 u7x3 + x5 + u3x9 + u4x10 + x17 + u6x18 1166 94 448 896
2.3 x3 + ux24 + x10 1166 96 896 896
2.4 x3 + u17(x17 + x18 + x20 + x24) 1168 96 64 64
2.5 x3 + u11x5 + u13x9 + x17 + u11x33 + x48 1170 96 320 320
2.6 u25x5 + x9 + u38x12 + u25x18 + u25x36 1170 96 64 64
2.7 u40x5 + u10x6 + u62x20 + u35x33 + u15x34 + u29x48 1170 96 64 64
2.8 u34x6 + u52x9 + u48x12 + u6x20 + u9x33 + u23x34 + u25x40 1170 96 64 64
2.9 x9 + u4(x10 + x18) + u9(x12 + x20 + x40) 1172 96 64 64
2.10 u52x3 + u47x5 + ux6 + u9x9 + u44x12 + u47x33 + u10x34 + u33x40 1172 96 64 64
2.11 u(x6 + x10 + x24 + x33) + x9 + u4x17 1174 96 64 64
2.12 x3 + u17(x17 + x18 + x20 + x24) + u14((u52x3 + u6x5 + u19x7 + u28x11 + u2x13)+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)2 + (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)4+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)8+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)16+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)32+ (u2x)9 +(u2x)19 +(u2x)36 + x21+x42 1300 152 8 8
[math]\displaystyle{ 7 }[/math] 1.1 [math]\displaystyle{ x^{3} }[/math] 3610 198 113792 113792
1.2 [math]\displaystyle{ x^{3} + {\rm Tr}(x^{9}) }[/math] 4026 212 896 896
2.1 [math]\displaystyle{ x^{34} + x^{18} + x^{5} }[/math] 4034 210 896 896
2.2 [math]\displaystyle{ x^{3} + x^{17} + x^{33} + x^{34} }[/math] 4040 212 896 896
3.1 [math]\displaystyle{ x^{5} }[/math] 3708 198 113792 113792
4.1 [math]\displaystyle{ x^{9} }[/math] 3610 198 113792 14565376
5.1 [math]\displaystyle{ x^{13} }[/math] 4270 338 889 889
6.1 [math]\displaystyle{ x^{57} }[/math] 4704 436 889 889
7.1 [math]\displaystyle{ x^{-1} }[/math] 8128 4928 1778 1778
8.1 [math]\displaystyle{ x^{65} + x^{10} + x^{3} }[/math] 4038 212 896 896
9.1 [math]\displaystyle{ x^{3} + x^{9} + x^{18} + x^{66} }[/math] 4044 212 896 896
10.1 [math]\displaystyle{ x^{3} + x^{12} + x^{17} + x^{33} }[/math] 4048 210 896 896
10.2 [math]\displaystyle{ x^{3} + x^{17} + x^{20} + x^{34} + x^{66} }[/math] 4040 210 896 896
11.1 [math]\displaystyle{ x^{3} + x^{20} + x^{34} + x^{66} }[/math] 4048 210 896 896
12.1 [math]\displaystyle{ x^{3} + x^{12} + x^{40} + x^{72} }[/math] 4048 210 896 896
13.1 [math]\displaystyle{ x^{3} + x^{5} + x^{10} + x^{33} + x^{34} }[/math] 4040 212 896 896
14.1 [math]\displaystyle{ x^{3} + x^{6} + x^{34} + x^{40} + x^{72} }[/math] 4048 212 896 896
14.2 [math]\displaystyle{ x^{3} + x^{5} + x^{6} + x^{12} + x^{33} + x^{34} }[/math] 4050 210 896 896
14.3 [math]\displaystyle{ u^{2}x^{96} + }[/math] [math]\displaystyle{ u^{78}x^{80} + }[/math] [math]\displaystyle{ u^{121}x^{72} + }[/math] [math]\displaystyle{ u^{49}x^{68} + }[/math] [math]\displaystyle{ u^{77}x^{66} + }[/math] [math]\displaystyle{ u^{29}x^{65} + }[/math] [math]\displaystyle{ u^{119}x^{48} + }[/math] [math]\displaystyle{ u^{117}x^{40} + }[/math] [math]\displaystyle{ u^{28}x^{36} + }[/math] [math]\displaystyle{ u^{107}x^{34} +u^{62}x^{33} +u^{125}x^{24} +u^{76}x^{20} +u^{84}x^{18} +u^{110}x^{17} +u^{49}x^{12} +u^{102}x^{10} +u^{69}x^{9} + }[/math] [math]\displaystyle{ u^{14}x^{6} + }[/math] [math]\displaystyle{ x^{5} + }[/math] [math]\displaystyle{ x^{3} }[/math] 4046 212 128 128
[math]\displaystyle{ 8 }[/math] 1.1 [math]\displaystyle{ x^{3} }[/math] 11818 420
1.2 [math]\displaystyle{ x^{9} }[/math] 12370 420
1.3 [math]\displaystyle{ x^{3}+{\rm Tr}(x^{9}) }[/math] 13800 432
1.4 [math]\displaystyle{ x^{9}+{\rm Tr}(x^{3}) }[/math] 13804 434
1.5 [math]\displaystyle{ x^{3}+u^{245}x^{33}+u^{183}x^{66}+u^{21}x^{144} }[/math] 13842 436
1.6 [math]\displaystyle{ x^{3} + u^{65}x^{18}+u^{120}x^{66}+u^{135}x^{144} }[/math] 13848 438
1.7 [math]\displaystyle{ u^{188}x^{192} + }[/math] [math]\displaystyle{ u^{129}x^{144} + }[/math] [math]\displaystyle{ u^{172}x^{132} + }[/math] [math]\displaystyle{ u^{138}x^{129} + }[/math] [math]\displaystyle{ u^{74}x^{96} + }[/math] [math]\displaystyle{ u^{244}x^{72} + }[/math] [math]\displaystyle{ u^{22}x^{66} + }[/math] [math]\displaystyle{ u^{178}x^{48} + }[/math] [math]\displaystyle{ u^{150}x^{36} + }[/math] [math]\displaystyle{ u^{146}x^{33} + }[/math] [math]\displaystyle{ u^{6}x^{24} + }[/math] [math]\displaystyle{ u^{60}x^{18} + }[/math] [math]\displaystyle{ u^{80}x^{12} + }[/math] [math]\displaystyle{ u^{140}x^{9} + }[/math] [math]\displaystyle{ u^{221}x^{6} + }[/math] [math]\displaystyle{ u^{19}x^{3} }[/math] 14034 438
1.8 [math]\displaystyle{ u^{37}x^{192} + }[/math] [math]\displaystyle{ u^{110}x^{144} + }[/math] [math]\displaystyle{ u^{40}x^{132} + }[/math] [math]\displaystyle{ u^{53}x^{129} + }[/math] [math]\displaystyle{ u^{239}x^{96} + }[/math] [math]\displaystyle{ u^{235}x^{72} + }[/math] [math]\displaystyle{ u^{126}x^{66} + }[/math] [math]\displaystyle{ u^{215}x^{48} + }[/math] [math]\displaystyle{ u^{96}x^{36} + }[/math] [math]\displaystyle{ u^{29}x^{33} + }[/math] [math]\displaystyle{ u^{19}x^{24} + }[/math] [math]\displaystyle{ u^{14}x^{18} + }[/math] [math]\displaystyle{ u^{139}x^{12} + }[/math] [math]\displaystyle{ u^{230}x^{9} + }[/math] [math]\displaystyle{ u^{234}x^{6} + }[/math] [math]\displaystyle{ u^{228}x^{3} }[/math] 14032 438
1.9 [math]\displaystyle{ u^{242}x^{192} + }[/math] [math]\displaystyle{ u^{100}x^{144} + }[/math] [math]\displaystyle{ u^{66}x^{132} + }[/math] [math]\displaystyle{ u^{230}x^{129} + }[/math] [math]\displaystyle{ u^{202}x^{96} + }[/math] [math]\displaystyle{ u^{156}x^{72} + }[/math] [math]\displaystyle{ u^{254}x^{66} + }[/math] [math]\displaystyle{ u^{18}x^{48} + }[/math] [math]\displaystyle{ u^{44}x^{36} + }[/math] [math]\displaystyle{ u^{95}x^{33} + }[/math] [math]\displaystyle{ u^{100}x^{24} + }[/math] [math]\displaystyle{ u^{245}x^{18} + }[/math] [math]\displaystyle{ u^{174}x^{12} + }[/math] [math]\displaystyle{ u^{175}x^{9} + }[/math] [math]\displaystyle{ u^{247}x^{6} + }[/math] [math]\displaystyle{ u^{166}x^{3} }[/math] 14036 438
1.10 [math]\displaystyle{ u^{100}x^{192} + }[/math] [math]\displaystyle{ u^{83}x^{144} + }[/math] [math]\displaystyle{ u^{153}x^{132} + }[/math] [math]\displaystyle{ u^{65}x^{129} + }[/math] [math]\displaystyle{ u^{174}x^{96} + }[/math] [math]\displaystyle{ u^{136}x^{72} + }[/math] [math]\displaystyle{ u^{46}x^{66}+ u^{55}x^{48}+ u^{224}x^{36}+ u^{180}x^{33}+ u^{179}x^{24}+u^{226}x^{18}+ u^{54}x^{12}+ u^{168}x^{9}+ u^{89}x^{6}+ u^{56}x^{3} }[/math] 14036 438
1.11 [math]\displaystyle{ u^{77}x^{192} + }[/math] [math]\displaystyle{ u^{133}x^{144} + }[/math] [math]\displaystyle{ u^{47}x^{132} + }[/math] [math]\displaystyle{ u^{229}x^{129} + }[/math] [math]\displaystyle{ u^{23}x^{96} + }[/math] [math]\displaystyle{ u^{242}x^{72} + }[/math] [math]\displaystyle{ u^{242}x^{66} + }[/math] [math]\displaystyle{ u^{245}x^{48} + }[/math] [math]\displaystyle{ u^{212}x^{36} + }[/math] [math]\displaystyle{ u^{231}x^{33} + }[/math] [math]\displaystyle{ u^{174}x^{24} + }[/math] [math]\displaystyle{ u^{216}x^{18} + }[/math] [math]\displaystyle{ u^{96}x^{12} + }[/math] [math]\displaystyle{ u^{253}x^{9} + }[/math] [math]\displaystyle{ u^{154}x^{6} + }[/math] [math]\displaystyle{ u^{71}x^{3} }[/math] 14032 438
1.12 [math]\displaystyle{ u^{220}x^{192} + }[/math] [math]\displaystyle{ u^{94}x^{144} + }[/math] [math]\displaystyle{ u^{70}x^{132} + }[/math] [math]\displaystyle{ u^{159}x^{129} + }[/math] [math]\displaystyle{ u^{145}x^{96} + }[/math] [math]\displaystyle{ u^{160}x^{72} + }[/math] [math]\displaystyle{ u^{74}x^{66} + }[/math] [math]\displaystyle{ u^{184}x^{48} + }[/math] [math]\displaystyle{ u^{119}x^{36} + }[/math] [math]\displaystyle{ u^{106}x^{33} + }[/math] [math]\displaystyle{ u^{253}x^{24} + }[/math] [math]\displaystyle{ wx^{18} + }[/math] [math]\displaystyle{ u^{90}x^{12} + }[/math] [math]\displaystyle{ u^{169}x^{9} + }[/math] [math]\displaystyle{ u^{118}x^{6} + }[/math] [math]\displaystyle{ u^{187}x^{3} }[/math] 14034 438
1.13 [math]\displaystyle{ u^{98}x^{192} + }[/math] [math]\displaystyle{ u^{225}x^{144} + }[/math] [math]\displaystyle{ u^{111}x^{132} + }[/math] [math]\displaystyle{ u^{238}x^{129} + }[/math] [math]\displaystyle{ u^{182}x^{96} + }[/math] [math]\displaystyle{ u^{125}x^{72} + }[/math] [math]\displaystyle{ u^{196}x^{66} + }[/math] [math]\displaystyle{ u^{219}x^{48} + }[/math] [math]\displaystyle{ u^{189}x^{36} + }[/math] [math]\displaystyle{ u^{199}x^{33} + }[/math] [math]\displaystyle{ u^{181}x^{24} + }[/math] [math]\displaystyle{ u^{110}x^{18} + }[/math] [math]\displaystyle{ u^{19}x^{12} + }[/math] [math]\displaystyle{ u^{175}x^{9} + }[/math] [math]\displaystyle{ u^{133}x^{6} + }[/math] [math]\displaystyle{ u^{47}x^{3} }[/math] 14030 438
1.14 [math]\displaystyle{ u^{236}x^{192} + }[/math] [math]\displaystyle{ u^{212}x^{160} + }[/math] [math]\displaystyle{ u^{153}x^{144} + }[/math] [math]\displaystyle{ u^{185}x^{136} + }[/math] [math]\displaystyle{ u^{3}x^{132} + }[/math] [math]\displaystyle{ u^{89}x^{130} + }[/math] [math]\displaystyle{ u^{189}x^{129} + }[/math] [math]\displaystyle{ u^{182}x^{96} + }[/math] [math]\displaystyle{ u^{105}x^{80} + }[/math] [math]\displaystyle{ u^{232}x^{72} + }[/math] [math]\displaystyle{ u^{219}x^{68} + }[/math] [math]\displaystyle{ u^{145}x^{66} + }[/math] [math]\displaystyle{ u^{171}x^{65} + }[/math] [math]\displaystyle{ u^{107}x^{48} + }[/math] [math]\displaystyle{ u^{179}x^{40} + }[/math] [math]\displaystyle{ u^{227}x^{36} + }[/math] [math]\displaystyle{ u^{236}x^{34} + }[/math] [math]\displaystyle{ u^{189}x^{33} + }[/math] [math]\displaystyle{ u^{162}x^{24} + }[/math] [math]\displaystyle{ u^{216}x^{20} + }[/math] [math]\displaystyle{ u^{162}x^{18} + }[/math] [math]\displaystyle{ u^{117}x^{17} + }[/math] [math]\displaystyle{ u^{56}x^{12} + }[/math] [math]\displaystyle{ u^{107}x^{10} + }[/math] [math]\displaystyle{ u^{236}x^{9} + }[/math] [math]\displaystyle{ u^{253}x^{6} + }[/math] [math]\displaystyle{ u^{180}x^{5} + }[/math] [math]\displaystyle{ u^{18}x^{3} }[/math] 14046 454
1.15 [math]\displaystyle{ u^{27}x^{192} + }[/math] [math]\displaystyle{ u^{167}x^{144} + }[/math] [math]\displaystyle{ u^{26}x^{132} + }[/math] [math]\displaystyle{ u^{231}x^{129} + }[/math] [math]\displaystyle{ u^{139}x^{96} + }[/math] [math]\displaystyle{ u^{30}x^{72} + }[/math] [math]\displaystyle{ u^{139}x^{66} + }[/math] [math]\displaystyle{ u^{203}x^{48} + }[/math] [math]\displaystyle{ u^{36}x^{36} + }[/math] [math]\displaystyle{ u^{210}x^{33} + }[/math] [math]\displaystyle{ u^{195}x^{24} + }[/math] [math]\displaystyle{ u^{12}x^{18} + }[/math] [math]\displaystyle{ u^{43}x^{12} + }[/math] [math]\displaystyle{ u^{97}x^{9} + }[/math] [math]\displaystyle{ u^{61}x^{6} + }[/math] [math]\displaystyle{ u^{39}x^{3} }[/math] 14036 454
1.16 [math]\displaystyle{ u^{6}x^{192} + }[/math] [math]\displaystyle{ u^{85}x^{144} + }[/math] [math]\displaystyle{ u^{251}x^{132} + }[/math] [math]\displaystyle{ u^{215}x^{129} + }[/math] [math]\displaystyle{ u^{229}x^{96} + }[/math] [math]\displaystyle{ u^{195}x^{72} + }[/math] [math]\displaystyle{ u^{152}x^{66} + }[/math] [math]\displaystyle{ u^{173}x^{48} + }[/math] [math]\displaystyle{ u^{209}x^{36} + }[/math] [math]\displaystyle{ u^{165}x^{33} + }[/math] [math]\displaystyle{ u^{213}x^{24} + }[/math] [math]\displaystyle{ u^{214}x^{18} + }[/math] [math]\displaystyle{ u^{158}x^{12} + }[/math] [math]\displaystyle{ u^{146}x^{9} + }[/math] [math]\displaystyle{ x^{6} + }[/math] [math]\displaystyle{ u^{50}x^{3} }[/math] 14032 438
1.17 [math]\displaystyle{ u^{164}x^{192} + }[/math] [math]\displaystyle{ u^{224}x^{144} + }[/math] [math]\displaystyle{ u^{59}x^{132} + }[/math] [math]\displaystyle{ u^{124}x^{129} + }[/math] [math]\displaystyle{ u^{207}x^{96} + }[/math] [math]\displaystyle{ u^{211}x^{72} + }[/math] [math]\displaystyle{ u^{5}x^{66} + }[/math] [math]\displaystyle{ u^{26}x^{48} + }[/math] [math]\displaystyle{ u^{20}x^{36} + }[/math] [math]\displaystyle{ u^{101}x^{33} + }[/math] [math]\displaystyle{ u^{175}x^{24} + }[/math] [math]\displaystyle{ u^{241}x^{18} + }[/math] [math]\displaystyle{ x^{12} + }[/math] [math]\displaystyle{ u^{15}x^{9} + }[/math] [math]\displaystyle{ u^{217}x^{6} + }[/math] [math]\displaystyle{ u^{212}x^{3} }[/math] 14028 438
2.1 [math]\displaystyle{ x^{3}+ x^{17}+u^{16}(x^{18}+x^{33})+u^{15}x^{48} }[/math] 13200 414
3.1 [math]\displaystyle{ x^{3}+ u^{24}x^{6}+u^{182}x^{132}+u^{67}x^{192} }[/math] 14024 438
4.1 [math]\displaystyle{ x^{3}+x^{6}+x^{68}+x^{80}+x^{132}+x^{160} }[/math] 14040 454
5.1 [math]\displaystyle{ x^{3}+x^{5}+x^{18}+x^{40}+x^{66} }[/math] 14044 446
6.1 [math]\displaystyle{ x^{3}+x^{12}+x^{40}+x^{66}+x^{130} }[/math] 14046 438
7.1 [math]\displaystyle{ x^{57} }[/math] 15358 960